Skip to main content
Log in

On the Minimum Consistent Subset Problem

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Let P be a set of n colored points in the d-dimensional Euclidean space. Introduced by Hart (1968), a consistent subset of P, is a set \(S\subseteq P\) such that for every point p in \(P {\setminus } S\), the closest point of p in S has the same color as p. The consistent subset problem is to find a consistent subset of P with minimum cardinality. This problem is known to be NP-complete even for two-colored point sets. Since the initial presentation of this problem, aside from the hardness results, there has not been significant progress from the algorithmic point of view. In this paper we present the following algorithmic results for the consistent subset problem in the plane: (1) The first subexponential-time algorithm for the consistent subset problem. (2) An \(O(n\log n)\)-time algorithm that finds a consistent subset of size two in two-colored point sets (if such a subset exists). Along the way we prove the following result which is of an independent interest: given n translations of a cone (defined as the intersection of n halfspaces) and n points in \(\mathbb {R}^3\), in \(O(n\log n)\) time one can decide whether or not there is a point in a cone. (3) An \(O(n\log ^2 n)\)-time algorithm that finds a minimum consistent subset in two-colored point sets where one color class contains exactly one point; this improves the previous best known \(O(n^2)\) running time which is due to Wilfong (SoCG 1991). (4) An O(n)-time algorithm for the consistent subset problem on collinear points that are given from left to right; this improves the previous best known \(O(n^2)\) running time. (5) A non-trivial \(O(n^6)\)-time dynamic programming algorithm for the consistent subset problem on points arranged on two parallel lines. To obtain these results, we combine tools from planar separators, paraboloid lifting, additively-weighted Voronoi diagrams with respect to convex distance functions, point location in farthest-point Voronoi diagrams, range trees, minimum covering of a circle with arcs, and several geometric transformations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. In some previous works the points have labels, as opposed to colors.

  2. The triangle is large in the sense that for every point \(p\in P\), the closest point to p, among \(P\cup \{v_1,v_2,v_3\}\), is in P.

  3. In fact the 2-connected 3-regular planar graph obtained from the Voronoi diagram of \(S\cup S'\) has such a separator.

  4. Wilfong shrinks the endpoint of \(A(b_i)\) that corresponds to \(cc(b_i)\) by half the clockwise angle from \(cc(b_i)\) to the next point, and shrinks the endpoint of \(A(b_i)\) that corresponds to \(c(b_i)\) by half the counterclockwise angle from \(c(b_i)\) to the previous point.

  5. The running time in [12, Lemma 3.15] has an extra logarithmic factor because they spend \(O(\log m)\) time to find the extremal vertex in a polygon M with m vertices.

References

  1. Banerjee, S., Bhore, S., Chitnis, R.: Algorithms and hardness results for nearest neighbor problems in bicolored point sets. In: Proceedings of the 13th Latin American Theoretical Informatics Symposium (LATIN), pp. 80–93 (2018)

  2. Bhattacharya, B.K., Bishnu, A., Cheong, O., Das, S., Karmakar, A., Snoeyink, J.: Computation of non-dominated points using compact Voronoi diagrams. In: Proceedings of the 4th International Workshop on Algorithms and Computation (WALCOM), pp. 82–93 (2010)

  3. de Berg, M., Khosravi, A.: Optimal binary space partitions for segments in the plane. Int. J. Comput. Geom. Appl. 22(3), 187–206 (2012)

    Article  MathSciNet  Google Scholar 

  4. Gates, G.: The reduced nearest neighbor rule. IEEE Trans. Inf. Theory 18(3), 431–433 (1972)

    Article  Google Scholar 

  5. Gottlieb, L., Kontorovich, A., Nisnevitch, P.: Near-optimal sample compression for nearest neighbors. IEEE Trans. Inf. Theory 64(6), 4120–4128 (2018). Also in NIPS 2014

    Article  MathSciNet  Google Scholar 

  6. Hart, P.E.: The condensed nearest neighbor rule. IEEE Trans. Inf. Theory 14(3), 515–516 (1968)

    Article  Google Scholar 

  7. Hwang, R.Z., Lee, R.C.T., Chang, R.C.: The slab dividing approach to solve the Euclidean \(p\)-center problem. Algorithmica 9(1), 1–22 (1993)

    Article  MathSciNet  Google Scholar 

  8. Khodamoradi, K., Krishnamurti, R., Roy, B.: Consistent subset problem with two labels. In: Proceedings of the 4th International Conference on Algorithms and Discrete Applied Mathematics (CALDAM), pp. 131–142 (2018)

  9. Kirkpatrick, D.G., Snoeyink, J.: Tentative prune-and-search for computing fixed-points with applications to geometric computation. Fundamenta Informaticae 22(4), 353–370 (1995). Also in SoCG 1993

    Article  MathSciNet  Google Scholar 

  10. Marx, D., Pilipczuk, M.: Optimal parameterized algorithms for planar facility location problems using voronoi diagrams. In: Proceedings of the 23rd Annual European Symposium on Algorithms (ESA), pp. 865–877 (2015). Full version in arXiv:1504.05476

  11. Masuyama, S., Ibaraki, T., Hasegawa, T.: Computational complexity of the \(m\)-center problems in the plane. Trans. Inst. Electron. Commun. Eng. Jpn. Sect. E E64(2), 57–64 (1981)

    Google Scholar 

  12. McAllister, M., Kirkpatrick, D.G., Snoeyink, J.: A compact piecewise-linear Voronoi diagram for convex sites in the plane. Discret. Comput. Geom. 15(1), 73–105 (1996). Also in FOCS 1993

    Article  MathSciNet  Google Scholar 

  13. Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. ACM 31(1), 114–127 (1984)

    Article  MathSciNet  Google Scholar 

  14. Miller, G.L.: Finding small simple cycle separators for 2-connected planar graphs. J. Comput. Syst. Sci. 32(3), 265–279 (1986). Also in STOC 1984

    Article  MathSciNet  Google Scholar 

  15. Ritter, G., Woodruff, H., Lowry, S., Isenhour, T.: An algorithm for a selective nearest neighbor decision rule. IEEE Trans. Inf. Theory 21(6), 665–669 (1975)

    Article  Google Scholar 

  16. Shamos, M.I., Hoey, D.: Closest-point problems. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science (FOCS), pp. 151–162 (1975)

  17. Wilfong, G.T.: Nearest neighbor problems. Int. J. Comput. Geom. Appl. 2(4), 383–416 (1992). Also in SoCG 1991

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work initiated at the Sixth Annual Workshop on Geometry and Graphs, March 11–16, 2018, at the Bellairs Research Institute of McGill University, Barbados. We are grateful to the organizers and to the participants of this workshop. We are also grateful to Otfried Cheong for helpful comments. We thank anonymous reviewers whose comments improved the readability of the paper. In particular the open problems mentioned in Sect. 7 are borrowed from these comments. Ahmad Biniaz was supported by NSERC Postdoctoral Fellowship. Sergio Cabello was supported by the Slovenian Research Agency, program P1-0297 and projects J1-8130, J1-8155. Paz Carmi was supported by grant 2016116 from the United States—Israel Binational Science Foundation. Jean-Lou De Carufel, Anil Maheshwari, and Michiel Smid were supported by NSERC. Saeed Mehrabi was supported by NSERC and by Carleton-Fields Postdoctoral Fellowship.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ahmad Biniaz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biniaz, A., Cabello, S., Carmi, P. et al. On the Minimum Consistent Subset Problem. Algorithmica 83, 2273–2302 (2021). https://doi.org/10.1007/s00453-021-00825-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-021-00825-8

Keywords

Navigation